Transcript Lecture Objectives - Jacksonville University

Lecture Objectives
1
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Learn how to think recursively
Trace a recursive method
Prove a recursive method is correct
Write recursive algorithms and methods for
searching arrays
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Recursion
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Recursion can solve many programming problems that are
difficult to solve linearly
In artificial intelligence, recursion is used to write programs that
 play games of chess
 prove mathematical theorems
 recognize patterns
Recursive algorithms can
 compute factorials
 compute a greatest common divisor
 process data structures (strings, arrays, linked lists, etc.)
 search efficiently using a binary search
 find a path through a maze, and more
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Recursive Thinking
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Recursive Thinking
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Recursion reduces a problem into one or more
simpler versions of itself
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Recursive Thinking (cont.)
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Recursive Thinking (cont.)
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Recursion of Process Nested Dreams
A child couldn't sleep, so her mother told a story about a little frog,
who couldn't sleep, so the frog's mother told a story about a little bear,
who couldn't sleep, so bear's mother told a story about a little weasel
...who fell asleep.
...and the little bear fell asleep;
...and the little frog fell asleep;
...and the child fell asleep.
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Recursive Thinking (cont.)
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Consider searching for a target value in an array
 With
elements sorted in increasing order
 Compare the target to the middle element
 If
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the middle element does not match the target
 search either the elements before the middle element
 or the elements after the middle element
Instead of searching n elements, we search n/2
elements
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Recursive Thinking (cont.)
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Recursive Algorithm to Search an Array
if the array is empty
return -1 as the search result
else if the middle element matches the target
return the subscript of the middle element as the result
else if the target is less than the middle element
recursively search the array elements before the middle
element and return the result
else
recursively search the array elements after the middle
element and return the result
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Steps to Design a Recursive
Algorithm
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Base case:
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Recursive case(s)
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for a small value of n, it can be solved directly
Smaller versions of the same problem
Algorithmic steps:
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Identify the base case and provide a solution to it
Reduce the problem to smaller versions of itself
Move towards the base case using smaller versions
Recursive Algorithm for Finding the
Length of a String
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if the string is empty (has no characters)
the length is 0
else
the length is 1 plus the length of the string that
excludes the first character
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Recursive Algorithm for Finding the
Length of a String (cont.)
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/** Recursive method length
@param str The string
@return The length of the string
*/
public static int length(String str) {
if (str == null || str.equals(""))
return 0;
else
return 1 + length(str.substring(1));
}
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Proving that a Recursive Method is
Correct
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Proof by induction
 Prove
the theorem base case is true
 Assuming that the theorem is true for n, prove it is true
for n+1
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Recursive proof is similar to induction. Verify that
 The
base case is recognized and solved correctly
 Each recursive case makes progress towards the base
case
 If all smaller problems are solved correctly, then the
original problem also is solved correctly
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Tracing a Recursive Method
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Unwinding the
recursion is the
process of
 returning
from
recursive calls
 computing the
partial results
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Run-Time Stack and Activation
Frames
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Java maintains a run-time stack
Activation frames save information in the run-time stack
The activation frame contains storage for
method arguments
 local variables (if any)
 the return address of the instruction that called the method
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Whenever a new method is called (recursive or not),
Java pushes a new activation frame into the run-time
stack
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Run-Time Stack and Activation
Frames (cont.)
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Run-Time Stack and Activation
Frames
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Recursive Algorithm for Printing
String Characters in Reverse
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/** Recursive method printCharsReverse
post: The argument string is displayed in reverse,
one character per line
@param str The string
*/
public static void printCharsReverse(String str) {
if (str == null || str.equals(""))
return;
else {
printCharsReverse(str.substring(1));
System.out.println(str.charAt(0));
}
}
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Recursive Algorithm for Finding
Duplicate characters
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/** Recursive method removeDuplicates
post: The word without duplicates
@param word The string
*/
public String removeDuplicates(String word) {
if(word == null || word.length() <= 1)
return word;
else if( word.charAt(0) == word.charAt(1) )
return removeDuplicates(word.substring(1,
word.length()));
else
return word.charAt(0) +
removeDuplicates(word.substring(1,
word.length()));
}
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Recursive Definitions of Mathematical Formulas
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Recursive Definitions of
Mathematical Formulas
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Mathematicians often use recursive definitions of
formulas
Examples include:
 factorials
 powers
 greatest
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common divisors (gcd)
Factorial of n: n!
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The factorial of n, or n! is defined as follows:
0! = 1
n! = n x (n -1)! (n > 0)
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The base case: n equal to 0
The second formula is a recursive definition
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Factorial of n: n! (cont.)
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The recursive definition can be expressed by the
following algorithm:
if n equals 0
n! is 1
else
n! = n x (n – 1)!

The last step can be implemented as:
return n * factorial(n – 1);
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Factorial of n: n! (cont.)
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public static int factorial(int n) {
if (n == 0)
return 1;
else
return n * factorial(n – 1);
}
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Infinite Recursion and Stack Overflow
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Call factorial with a negative argument, what
will happen?
StackOverflowException
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Recursive Algorithm for Calculating
xn
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Eclipse example
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Recursive Algorithm for Calculating
gcd
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The greatest common divisor (gcd) of two numbers
is the largest integer that divides both numbers
The gcd of 20 and 15 is 5
The gcd of 36 and 24 is 12
The gcd of 38 and 18 is 2
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Recursive Algorithm for Calculating
gcd (cont.)
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Given 2 positive integers m and n (m > n)
if n is a divisor of m
gcd(m, n) = n
else
gcd (m, n) = gcd (n, m % n)
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Recursive Algorithm for Calculating
gcd (cont.)
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/** Recursive gcd method (in RecursiveMethods.java).
pre: m > 0 and n > 0
@param m The larger number
@param n The smaller number
@return Greatest common divisor of m and n
*/
public static double gcd(int m, int n) {
if (m % n == 0)
return n;
else if (m < n)
return gcd(n, m); // Transpose arguments.
else
return gcd(n, m % n);
}
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Recursion Versus Iteration
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Similarities between recursion and iteration
Iteration
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Recursion
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Loop repetition condition determines whether to repeat the
loop body or
exit from the loop
The condition usually tests for a base case
You can always write an iterative solution to a recursion
But can you always write a recursion for an iteration
problem?
Which one is best?
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Recursion Versus Iteration
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Difference between iteration and recursion is:
with iteration, each step clearly leads onto the next,
like stepping stones across a river,
 in recursion, each step replicates itself at a smaller
scale, so that all of them combined together
eventually solve the problem.

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Iterative factorial Method
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/** Iterative factorial method.
pre: n >= 0
@param n The integer whose factorial is being computed
@return n!
*/
public static int factorialIter(int n) {
int result = 1;
for (int k = 1; k <= n; k++)
result = result * k;
return result;
}
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Efficiency of Recursion
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Recursive methods often have slower execution times
 Why?
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What about memory and recursion vs iteration?
Why would we ever prefer recursion?
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Fibonacci Numbers
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Fibonacci numbers were invented to model the
growth of a rabbit colony
fib1 = 1
fib2 = 1
fibn = fibn-1 + fibn-2
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
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An Exponential Recursive
fibonacci Method
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Efficiency of Recursion: Exponential
fibonacci
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Inefficient
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An O(n) Recursive fibonacci
Method
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An O(n) Recursive fibonacci
Method (cont.)
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Non-recursive wrapper method:
/** Wrapper method for calculating Fibonacci numbers
(in RecursiveMethods.java).
pre: n >= 1
@param n The position of the desired Fibonacci
number
@return The value of the nth Fibonacci number
*/
public static int fibonacciStart(int n) {
return fibo(1, 0, n);
}
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Efficiency of Recursion: O(n)
fibonacci
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Efficient
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Efficiency of Recursion: O(n)
fibonacci
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Method fibo is an example of tail recursion or lastline recursion
Why is it more effective?
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Recursive Array Search
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Recursive Array Search
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Simplest way to search is a linear search
 Start
with the first element
 Examine one element at a time
 End with the last
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(n + 1)/2 elements examined in a linear search
 If
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the target is not in the list, n elements are examined
What is the complexity?
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Recursive Array Search (cont.)
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Base cases for recursive search:
 Empty
array, target can not be found; result is -1
 First element of the array being searched = target?
 Yes? Then result is the subscript of first element
 No? Then recursive step searches the rest of the array,
excluding the first element
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Algorithm for Recursive Linear Array
Search
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if the array is empty
the result is –1
else if the first element matches the target
the result is the subscript of the first element
else
search the array excluding the first element and return
the result
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Implementation of Recursive Linear
Search
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Implementation of Recursive Linear
Search (cont.)
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
A non-recursive wrapper method:
/** Wrapper for recursive linear search method
@param items The array being searched
@param target The object being searched for
@return The subscript of target if found;
otherwise -1
*/
public static int linearSearch(Object[] items, Object target)
{
return linearSearch(items, target, 0);
}
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Implementation of Recursive Linear
Search (cont.)
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Design of a Binary Search Algorithm
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Array has to be sorted
Base cases
 The
array is empty
 Element being examined matches the target
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Compares the middle element for a match with the
target
Excludes the half of the array within which the
target cannot lie
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Binary Search Algorithm (cont.)
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if the array is empty
return –1 as the search result
else if the middle element matches the target
return the subscript of the middle element as the result
else if the target is less than the middle element
recursively search the array elements before the middle
element
and return the result
else
recursively search the array elements after the middle
element and
return the result
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Binary Search Algorithm
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target
First call
England
China
first = 0
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Denmark
England
Greece
middle = 3
Japan
Russia
USA
last = 6
Binary Search Algorithm (cont.)
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target
Second call
England
China
Denmark
first = 0
last = 2
middle = 1
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England
Greece
Japan
Russia
USA
Binary Search Algorithm (cont.)
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target
Third call
England
China
Denmark
England
Greece
first= middle = last = 2
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Japan
Russia
USA
Efficiency of Binary Search
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An array of 16 would search arrays of length 16, 8,
4, 2, and 1; 5 probes in the worst case
= 24
 5 = log216 + 1
 16
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A doubled array size would only require 6 probes
in the worst case
= 25
 6 = log232 + 1
 32
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An array with 32,768 elements requires only 16
probes! (log232768 = 15)
Comparable Interface
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Comparable interface classes must define a
compareTo method
Method obj1.compareTo(obj2) returns an
integer
 negative:
obj1 < obj2
 zero: obj1 == obj2
 positive: obj1 > obj2
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Implementation of a Binary Search
Algorithm
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Implementation of a Binary Search
Algorithm (cont.)
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Trace of Binary Search
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Testing Binary Search
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You should test arrays with
 an
even number of elements
 an odd number of elements
 duplicate elements
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Test each array for the following cases:
 the
target is the element at each position of the array
 the target is less than the smallest array element
 the target is greater than the largest array element
 the target is a value between each pair of items in the
array
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Method Arrays.binarySearch
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Java API class Arrays contains a binarySearch
method
 If
the objects in the array are not comparable or if the
array is not sorted, the results are undefined
 If there are multiple copies of the target value in the
array, there is no guarantee which one will be found
 Throws ClassCastException if the target is not
comparable to the array elements
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